Pedrocchi, Fabio Luigi. Spin systems and longrange interactions for quantum memories and quantum computing. 2013, Doctoral Thesis, University of Basel, Faculty of Science.

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Abstract
Since the seminal work by Shor who proposed a quantum algorithm factorizing integers into prime factors, it has become manifest that the laws of quantum mechanics provide resources for computation that overpower classical physics. The computational advantages that quantum physics offers have stimulated a tremendous amount of theoretical and experimental research. In this context, spin systems have played a major role, given that the spin degree of freedom  with the paradigmatic case of the spin1/2 of electrons  represents an obvious candidate for the encoding of an elementary bit of quantum information (qubit).
On the other hand, however, quantum objects are very fragile entities, being very susceptible to the environment they reside in. This fragility of qubits is one of the main obstacles in the realization of a quantum computer.
In this thesis, we mainly address the two following questions relevant to quantum computation.
i) How is it possible to realize quantum gates both in a reliable and scalable way?
ii) How can we store quantum information in a way that is resilient to the errors caused by the thermal environment?
We focus on spin systems and demonstrate that longrange spinspin interactions in the models considered can have beneficial effects.
In their pioneering work, Loss and DiVincenzo proposed a way to perform quantum computation in a semiconductorbased architecture where the spin state of an electron trapped in a quantum dot is chosen to encode the elementary qubit. In this proposal, the spins are required to lie spatially close to each other, and this might complicate the realization of a scalable architecture.
In the first part of the thesis we thus propose a scheme that allows the constraint on the positioning of the qubits to be relaxed. This is achieved by introducing a ferromagnetic coupler between the distant qubits, to which it is coupled via a dipolar interaction. Most importantly, our proposal is applicable to any type of spin qubits and in particular to the technologically very relevant siliconbased qubits and NVcenters in diamond to which previous coupling schemes do not apply.
As additional key element, a quantum computer needs a memory capable of reliably storing quantum information in the presence of thermal fluctuations.
This brings us to the second part of this thesis, where we consider selfcorrecting memories, for which the protection against thermal noise is builtin at the hardware level. We propose physical models that exhibit these selfcorrecting properties, using as a starting point the well known topologically ordered toric code. In particular, we investigate how to induce longrange interactions between the spins of the toric code, since such interactions help increase the memory lifetime.
As a first step, we study a honeycomb quantum spin model coupled to delocalized cavity modes. We investigate the properties of the lowenergy toric code Hamiltonian and show that the coupling to cavity modes prolongs the lifetime of the memory and offers a method to detect the presence of excitations.
While the introduction of extended bosonic modes makes the model nonlocal, we also propose a purely local model consisting of a toric code embedded in a threedimensional cubic lattice of hopping bosons; the lowenergy sector of a toric code coupled to a threedimensional Heisenberg ferromagnet in a brokensymmetry state realizes this model. Our analysis leads to an energy penalty for the creation of defects that grows linearly with the size L of the memory and thus to a lifetime increasing exponentially with L.
In the third part of this thesis, we study spin systems that support anyons, i.e., particles with fractional statistics. Similar to the toric code, such systems are topologically ordered: they are immune to local perturbations and quantum gates are implemented by nonlocal operations, namely the exchange of anyons, whose outcomes depend only on the topology of the exchange. Here again the faulttolerance is achieved at the level of the hardware and physical systems supporting nonabelian anyons are thus promising platforms for quantum computation.
We focus on spin systems that exhibit some of these properties and specifically on variations of the honeycomb quantum spin model. We first investigate the exact solution of the honeycomb model in detail and derive an explicit formula for the projector onto the physical subspace. We use this result to study inhomogeneous open spin ladders, related to the honeycomb model, which can be tuned between topological and nontopological phases. We test the robustness of Majorana end states (MES) which emerge at the boundary between sections in different topological phases. Furthermore, we present a trijunction setup where MES can be braided. This is of interest since MES in these spin ladders potentially follow nonabelian braiding statistics. Finally, we study the ground states of the aforementioned ladders and show that they are free of vortices when the signs of the spin couplings are all positive or negative. To prove this, we use exact reflectionpositivitybased methods as well as approximate methods.
In the last part of the thesis, we provide an extension of the MerminWagner theorem to a system of lattice spins that are spincoupled to itinerant and interacting charge carriers. We prove that neither (anti) ferromagnetic nor helical longrange order is possible in one and two dimensions at any finite temperature (in the absence of spinorbit). The fundamental question whether spontaneous ordering of the lattice spins occurs in these systems is of interest in the context of quantum computation; the polarization of nuclear spins coupled to a twodimensional electron gas is a possible route towards the reduction of decoherence induced by the fluctuating Overhauser field in gatedefined quantum dots.
On the other hand, however, quantum objects are very fragile entities, being very susceptible to the environment they reside in. This fragility of qubits is one of the main obstacles in the realization of a quantum computer.
In this thesis, we mainly address the two following questions relevant to quantum computation.
i) How is it possible to realize quantum gates both in a reliable and scalable way?
ii) How can we store quantum information in a way that is resilient to the errors caused by the thermal environment?
We focus on spin systems and demonstrate that longrange spinspin interactions in the models considered can have beneficial effects.
In their pioneering work, Loss and DiVincenzo proposed a way to perform quantum computation in a semiconductorbased architecture where the spin state of an electron trapped in a quantum dot is chosen to encode the elementary qubit. In this proposal, the spins are required to lie spatially close to each other, and this might complicate the realization of a scalable architecture.
In the first part of the thesis we thus propose a scheme that allows the constraint on the positioning of the qubits to be relaxed. This is achieved by introducing a ferromagnetic coupler between the distant qubits, to which it is coupled via a dipolar interaction. Most importantly, our proposal is applicable to any type of spin qubits and in particular to the technologically very relevant siliconbased qubits and NVcenters in diamond to which previous coupling schemes do not apply.
As additional key element, a quantum computer needs a memory capable of reliably storing quantum information in the presence of thermal fluctuations.
This brings us to the second part of this thesis, where we consider selfcorrecting memories, for which the protection against thermal noise is builtin at the hardware level. We propose physical models that exhibit these selfcorrecting properties, using as a starting point the well known topologically ordered toric code. In particular, we investigate how to induce longrange interactions between the spins of the toric code, since such interactions help increase the memory lifetime.
As a first step, we study a honeycomb quantum spin model coupled to delocalized cavity modes. We investigate the properties of the lowenergy toric code Hamiltonian and show that the coupling to cavity modes prolongs the lifetime of the memory and offers a method to detect the presence of excitations.
While the introduction of extended bosonic modes makes the model nonlocal, we also propose a purely local model consisting of a toric code embedded in a threedimensional cubic lattice of hopping bosons; the lowenergy sector of a toric code coupled to a threedimensional Heisenberg ferromagnet in a brokensymmetry state realizes this model. Our analysis leads to an energy penalty for the creation of defects that grows linearly with the size L of the memory and thus to a lifetime increasing exponentially with L.
In the third part of this thesis, we study spin systems that support anyons, i.e., particles with fractional statistics. Similar to the toric code, such systems are topologically ordered: they are immune to local perturbations and quantum gates are implemented by nonlocal operations, namely the exchange of anyons, whose outcomes depend only on the topology of the exchange. Here again the faulttolerance is achieved at the level of the hardware and physical systems supporting nonabelian anyons are thus promising platforms for quantum computation.
We focus on spin systems that exhibit some of these properties and specifically on variations of the honeycomb quantum spin model. We first investigate the exact solution of the honeycomb model in detail and derive an explicit formula for the projector onto the physical subspace. We use this result to study inhomogeneous open spin ladders, related to the honeycomb model, which can be tuned between topological and nontopological phases. We test the robustness of Majorana end states (MES) which emerge at the boundary between sections in different topological phases. Furthermore, we present a trijunction setup where MES can be braided. This is of interest since MES in these spin ladders potentially follow nonabelian braiding statistics. Finally, we study the ground states of the aforementioned ladders and show that they are free of vortices when the signs of the spin couplings are all positive or negative. To prove this, we use exact reflectionpositivitybased methods as well as approximate methods.
In the last part of the thesis, we provide an extension of the MerminWagner theorem to a system of lattice spins that are spincoupled to itinerant and interacting charge carriers. We prove that neither (anti) ferromagnetic nor helical longrange order is possible in one and two dimensions at any finite temperature (in the absence of spinorbit). The fundamental question whether spontaneous ordering of the lattice spins occurs in these systems is of interest in the context of quantum computation; the polarization of nuclear spins coupled to a twodimensional electron gas is a possible route towards the reduction of decoherence induced by the fluctuating Overhauser field in gatedefined quantum dots.
Advisors:  Loss, Daniel 

Committee Members:  Di Vincenzo, David P. 
Faculties and Departments:  05 Faculty of Science > Departement Physik > Physik > Theoretische Physik Mesoscopics (Loss) 
UniBasel Contributors:  Pedrocchi, Fabio Luigi and Loss, Daniel 
Item Type:  Thesis 
Thesis Subtype:  Doctoral Thesis 
Thesis no:  10592 
Thesis status:  Complete 
Number of Pages:  319 S. 
Language:  English 
Identification Number: 

edoc DOI:  
Last Modified:  22 Jan 2018 15:51 
Deposited On:  06 Dec 2013 13:43 
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